Abstract

The extended Jones formulation is used to investigate propagation at nonnormal incidence through two- and three-layer systems of birefringent material in which the optic axes of the individual layers are in the plane of the layers. Such systems are equivalent to two optical elements in series—an equivalent retardation plate and a polarization rotator. Analytical solutions are obtained for the equivalent retardation and rotation. The major finding is that, in general, there are two nonnormal incidence directions for which the retardation vanishes; therefore these two directions are optic axes of the composite system. These simple layered systems therefore behave in a manner similar to biaxial crystals. Moreover, the results illustrate the fact that even if the optic axes of individual layers in composite systems are in the plane of the layers, the optic axes of the system are, in general, out of this plane.

© 2005 Optical Society of America

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References

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  1. D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “Numerical modeling of the cornea’s lamellar structure and birefringence properties,” J. Opt. Soc. Am. A 12, 1425–1438 (1995).
    [CrossRef]
  2. D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “A numerical test of the normal incidence uniaxial model of corneal birefringence,” Cornea 5, 278–285 (1996).
    [CrossRef]
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    [CrossRef]
  4. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
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  5. R. C. Jones, “A new calculus for the treatment of optical systems III. The Sohncke theory of optical activity,” J. Opt. Soc. Am. 31, 500–503 (1941).
    [CrossRef]
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    [CrossRef]
  7. R. A. Farrell, J. F. Wharam, D. Kim, R. L. McCally, “Polarized light propagation in corneal lamellae,” J. Refract. Surg. 15, 700–705 (1999).
    [PubMed]
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    [CrossRef]
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    [CrossRef]
  10. L. J. Bour, N. J. Lopes Cardozo, “On the birefringence of the living human eye,” Vision Res. 21, 1413–1421 (1981).
    [CrossRef] [PubMed]
  11. A. Stanworth, E. J. Naylor, “The polarization optics of the isolated cornea,” Br. J. Ophthamol. 34, 201–211 (1950).
    [CrossRef]
  12. A. Stanworth, E. J. Naylor, “Polarized light studies of the cornea. I. The isolated cornea,” J. Exp. Biol. 30, 160–163 (1953).
  13. G. Boehm, “Über Maculare (Haidingersche) Polarisations Büschel und Über einen Polarisation Optischen Fehler des Auges,” Acta Ophthalmol. 18, 109–142 (1940).
    [CrossRef]
  14. C. C. D. Shute, “Haidinger’s brushes and predominant orientation of collagen in corneal stroma,” Nature 250, 163–164 (1974).
    [CrossRef] [PubMed]
  15. H. L. D. Vries, A. Spoor, R. Gielof, “Properties of the eye with respect to polarized light,” Physica (Amsterdam) 19, 419–432 (1953).
    [CrossRef]
  16. G. J. van Blokland, S. C. Verhelst, “Corneal polarization in the living human eye explained with a biaxial model,” J. Opt. Soc. Am. A 4, 82–90 (1987).
    [CrossRef] [PubMed]
  17. D. M. Maurice, “The structure and transparency of the corneal stroma,” J. Physiol. (London) 136, 263–285 (1957).
  18. D. S. Greenfield, R. W. Knighton, X.-R. Huang, “Effect of polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000).
    [CrossRef] [PubMed]
  19. R. W. Knighton, X.-R. Huang, “Corneal compensation in scanning laser polarimetry: characterization and analysis,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S92 (2000).
  20. R. W. Knighton, X.-R. Huang, D. S. Greenfield, “Linear birerefringence measured in the central corneas of a normal population,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S131 (2001).
  21. R. A. Farrell, R. L. McCally, D. Rouseff, “Corneal birefringence models,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S281 (2001).
  22. D. Rouseff, R. A. Farrell, R. L. McCally, “Numerical modeling of the cornea’s birefringence properties. Nonnormal incidence effects,” manuscript available from the authors (russell.mccally@jhuap/.edu).
  23. As used by Jones, S(ϕ) has the effect of rotating a vector counterclockwise through an angle ϕ. S(−ϕ) is the standard rotation matrix used to express a vector in a coordinate system rotated counterclockwise by an angle ϕ from the “laboratory system.”
  24. This can be shown rigorously by examining the implications of setting θ=0 in Eq. (19). Doing so, and realizing that both of the squared terms must be zero in order for γ¯ to be zero, we obtain two equations for tan2ϕr as functions of ϕ3. We used Mathematica to show that the two equations do not have a common solution.
  25. W. A. Christens-Barry, W. J. Green, P. J. Connolly, R. A. Farrell, R. L. McCally, “Spatial mapping of polarized light transmission in the central rabbit cornea,” Exp. Eye Res. 62, 651–662 (1996).
    [CrossRef] [PubMed]
  26. J. W. Jaronski, H. T. Kasprzak, “Linear birefringence measurements of the in vitro human cornea,” Ophthalmic Physiol. Opt. 23, 361–369 (2003).
    [CrossRef] [PubMed]
  27. R. W. Knighton, X.-R. Huang, “Linear birefringence of the central human cornea,” Invest. Ophthalmol. Visual Sci. 43, 82–86 (2002).

2003 (1)

J. W. Jaronski, H. T. Kasprzak, “Linear birefringence measurements of the in vitro human cornea,” Ophthalmic Physiol. Opt. 23, 361–369 (2003).
[CrossRef] [PubMed]

2002 (1)

R. W. Knighton, X.-R. Huang, “Linear birefringence of the central human cornea,” Invest. Ophthalmol. Visual Sci. 43, 82–86 (2002).

2001 (2)

R. W. Knighton, X.-R. Huang, D. S. Greenfield, “Linear birerefringence measured in the central corneas of a normal population,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S131 (2001).

R. A. Farrell, R. L. McCally, D. Rouseff, “Corneal birefringence models,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S281 (2001).

2000 (2)

D. S. Greenfield, R. W. Knighton, X.-R. Huang, “Effect of polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000).
[CrossRef] [PubMed]

R. W. Knighton, X.-R. Huang, “Corneal compensation in scanning laser polarimetry: characterization and analysis,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S92 (2000).

1999 (1)

R. A. Farrell, J. F. Wharam, D. Kim, R. L. McCally, “Polarized light propagation in corneal lamellae,” J. Refract. Surg. 15, 700–705 (1999).
[PubMed]

1996 (2)

W. A. Christens-Barry, W. J. Green, P. J. Connolly, R. A. Farrell, R. L. McCally, “Spatial mapping of polarized light transmission in the central rabbit cornea,” Exp. Eye Res. 62, 651–662 (1996).
[CrossRef] [PubMed]

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “A numerical test of the normal incidence uniaxial model of corneal birefringence,” Cornea 5, 278–285 (1996).
[CrossRef]

1995 (1)

1993 (1)

1987 (1)

1982 (1)

1981 (1)

L. J. Bour, N. J. Lopes Cardozo, “On the birefringence of the living human eye,” Vision Res. 21, 1413–1421 (1981).
[CrossRef] [PubMed]

1974 (1)

C. C. D. Shute, “Haidinger’s brushes and predominant orientation of collagen in corneal stroma,” Nature 250, 163–164 (1974).
[CrossRef] [PubMed]

1957 (1)

D. M. Maurice, “The structure and transparency of the corneal stroma,” J. Physiol. (London) 136, 263–285 (1957).

1953 (2)

H. L. D. Vries, A. Spoor, R. Gielof, “Properties of the eye with respect to polarized light,” Physica (Amsterdam) 19, 419–432 (1953).
[CrossRef]

A. Stanworth, E. J. Naylor, “Polarized light studies of the cornea. I. The isolated cornea,” J. Exp. Biol. 30, 160–163 (1953).

1950 (1)

A. Stanworth, E. J. Naylor, “The polarization optics of the isolated cornea,” Br. J. Ophthamol. 34, 201–211 (1950).
[CrossRef]

1942 (1)

1941 (3)

1940 (1)

G. Boehm, “Über Maculare (Haidingersche) Polarisations Büschel und Über einen Polarisation Optischen Fehler des Auges,” Acta Ophthalmol. 18, 109–142 (1940).
[CrossRef]

Boehm, G.

G. Boehm, “Über Maculare (Haidingersche) Polarisations Büschel und Über einen Polarisation Optischen Fehler des Auges,” Acta Ophthalmol. 18, 109–142 (1940).
[CrossRef]

Bour, L. J.

L. J. Bour, N. J. Lopes Cardozo, “On the birefringence of the living human eye,” Vision Res. 21, 1413–1421 (1981).
[CrossRef] [PubMed]

Christens-Barry, W. A.

W. A. Christens-Barry, W. J. Green, P. J. Connolly, R. A. Farrell, R. L. McCally, “Spatial mapping of polarized light transmission in the central rabbit cornea,” Exp. Eye Res. 62, 651–662 (1996).
[CrossRef] [PubMed]

Connolly, P. J.

W. A. Christens-Barry, W. J. Green, P. J. Connolly, R. A. Farrell, R. L. McCally, “Spatial mapping of polarized light transmission in the central rabbit cornea,” Exp. Eye Res. 62, 651–662 (1996).
[CrossRef] [PubMed]

Donohue, D. J.

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “A numerical test of the normal incidence uniaxial model of corneal birefringence,” Cornea 5, 278–285 (1996).
[CrossRef]

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “Numerical modeling of the cornea’s lamellar structure and birefringence properties,” J. Opt. Soc. Am. A 12, 1425–1438 (1995).
[CrossRef]

Farrell, R. A.

R. A. Farrell, R. L. McCally, D. Rouseff, “Corneal birefringence models,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S281 (2001).

R. A. Farrell, J. F. Wharam, D. Kim, R. L. McCally, “Polarized light propagation in corneal lamellae,” J. Refract. Surg. 15, 700–705 (1999).
[PubMed]

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “A numerical test of the normal incidence uniaxial model of corneal birefringence,” Cornea 5, 278–285 (1996).
[CrossRef]

W. A. Christens-Barry, W. J. Green, P. J. Connolly, R. A. Farrell, R. L. McCally, “Spatial mapping of polarized light transmission in the central rabbit cornea,” Exp. Eye Res. 62, 651–662 (1996).
[CrossRef] [PubMed]

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “Numerical modeling of the cornea’s lamellar structure and birefringence properties,” J. Opt. Soc. Am. A 12, 1425–1438 (1995).
[CrossRef]

D. Rouseff, R. A. Farrell, R. L. McCally, “Numerical modeling of the cornea’s birefringence properties. Nonnormal incidence effects,” manuscript available from the authors (russell.mccally@jhuap/.edu).

Gielof, R.

H. L. D. Vries, A. Spoor, R. Gielof, “Properties of the eye with respect to polarized light,” Physica (Amsterdam) 19, 419–432 (1953).
[CrossRef]

Green, W. J.

W. A. Christens-Barry, W. J. Green, P. J. Connolly, R. A. Farrell, R. L. McCally, “Spatial mapping of polarized light transmission in the central rabbit cornea,” Exp. Eye Res. 62, 651–662 (1996).
[CrossRef] [PubMed]

Greenfield, D. S.

R. W. Knighton, X.-R. Huang, D. S. Greenfield, “Linear birerefringence measured in the central corneas of a normal population,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S131 (2001).

D. S. Greenfield, R. W. Knighton, X.-R. Huang, “Effect of polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000).
[CrossRef] [PubMed]

Gu, C.

Huang, X.-R.

R. W. Knighton, X.-R. Huang, “Linear birefringence of the central human cornea,” Invest. Ophthalmol. Visual Sci. 43, 82–86 (2002).

R. W. Knighton, X.-R. Huang, D. S. Greenfield, “Linear birerefringence measured in the central corneas of a normal population,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S131 (2001).

D. S. Greenfield, R. W. Knighton, X.-R. Huang, “Effect of polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000).
[CrossRef] [PubMed]

R. W. Knighton, X.-R. Huang, “Corneal compensation in scanning laser polarimetry: characterization and analysis,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S92 (2000).

Hurwitz, H.

Jaronski, J. W.

J. W. Jaronski, H. T. Kasprzak, “Linear birefringence measurements of the in vitro human cornea,” Ophthalmic Physiol. Opt. 23, 361–369 (2003).
[CrossRef] [PubMed]

Jones, R. C.

Kasprzak, H. T.

J. W. Jaronski, H. T. Kasprzak, “Linear birefringence measurements of the in vitro human cornea,” Ophthalmic Physiol. Opt. 23, 361–369 (2003).
[CrossRef] [PubMed]

Kim, D.

R. A. Farrell, J. F. Wharam, D. Kim, R. L. McCally, “Polarized light propagation in corneal lamellae,” J. Refract. Surg. 15, 700–705 (1999).
[PubMed]

Knighton, R. W.

R. W. Knighton, X.-R. Huang, “Linear birefringence of the central human cornea,” Invest. Ophthalmol. Visual Sci. 43, 82–86 (2002).

R. W. Knighton, X.-R. Huang, D. S. Greenfield, “Linear birerefringence measured in the central corneas of a normal population,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S131 (2001).

R. W. Knighton, X.-R. Huang, “Corneal compensation in scanning laser polarimetry: characterization and analysis,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S92 (2000).

D. S. Greenfield, R. W. Knighton, X.-R. Huang, “Effect of polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000).
[CrossRef] [PubMed]

Lopes Cardozo, N. J.

L. J. Bour, N. J. Lopes Cardozo, “On the birefringence of the living human eye,” Vision Res. 21, 1413–1421 (1981).
[CrossRef] [PubMed]

Maurice, D. M.

D. M. Maurice, “The structure and transparency of the corneal stroma,” J. Physiol. (London) 136, 263–285 (1957).

McCally, R. L.

R. A. Farrell, R. L. McCally, D. Rouseff, “Corneal birefringence models,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S281 (2001).

R. A. Farrell, J. F. Wharam, D. Kim, R. L. McCally, “Polarized light propagation in corneal lamellae,” J. Refract. Surg. 15, 700–705 (1999).
[PubMed]

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “A numerical test of the normal incidence uniaxial model of corneal birefringence,” Cornea 5, 278–285 (1996).
[CrossRef]

W. A. Christens-Barry, W. J. Green, P. J. Connolly, R. A. Farrell, R. L. McCally, “Spatial mapping of polarized light transmission in the central rabbit cornea,” Exp. Eye Res. 62, 651–662 (1996).
[CrossRef] [PubMed]

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “Numerical modeling of the cornea’s lamellar structure and birefringence properties,” J. Opt. Soc. Am. A 12, 1425–1438 (1995).
[CrossRef]

D. Rouseff, R. A. Farrell, R. L. McCally, “Numerical modeling of the cornea’s birefringence properties. Nonnormal incidence effects,” manuscript available from the authors (russell.mccally@jhuap/.edu).

Naylor, E. J.

A. Stanworth, E. J. Naylor, “Polarized light studies of the cornea. I. The isolated cornea,” J. Exp. Biol. 30, 160–163 (1953).

A. Stanworth, E. J. Naylor, “The polarization optics of the isolated cornea,” Br. J. Ophthamol. 34, 201–211 (1950).
[CrossRef]

Rouseff, D.

R. A. Farrell, R. L. McCally, D. Rouseff, “Corneal birefringence models,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S281 (2001).

D. Rouseff, R. A. Farrell, R. L. McCally, “Numerical modeling of the cornea’s birefringence properties. Nonnormal incidence effects,” manuscript available from the authors (russell.mccally@jhuap/.edu).

Shute, C. C. D.

C. C. D. Shute, “Haidinger’s brushes and predominant orientation of collagen in corneal stroma,” Nature 250, 163–164 (1974).
[CrossRef] [PubMed]

Spoor, A.

H. L. D. Vries, A. Spoor, R. Gielof, “Properties of the eye with respect to polarized light,” Physica (Amsterdam) 19, 419–432 (1953).
[CrossRef]

Stanworth, A.

A. Stanworth, E. J. Naylor, “Polarized light studies of the cornea. I. The isolated cornea,” J. Exp. Biol. 30, 160–163 (1953).

A. Stanworth, E. J. Naylor, “The polarization optics of the isolated cornea,” Br. J. Ophthamol. 34, 201–211 (1950).
[CrossRef]

Stoyanov, B. J.

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “A numerical test of the normal incidence uniaxial model of corneal birefringence,” Cornea 5, 278–285 (1996).
[CrossRef]

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “Numerical modeling of the cornea’s lamellar structure and birefringence properties,” J. Opt. Soc. Am. A 12, 1425–1438 (1995).
[CrossRef]

van Blokland, G. J.

Verhelst, S. C.

Vries, H. L. D.

H. L. D. Vries, A. Spoor, R. Gielof, “Properties of the eye with respect to polarized light,” Physica (Amsterdam) 19, 419–432 (1953).
[CrossRef]

Wharam, J. F.

R. A. Farrell, J. F. Wharam, D. Kim, R. L. McCally, “Polarized light propagation in corneal lamellae,” J. Refract. Surg. 15, 700–705 (1999).
[PubMed]

Yeh, P.

Acta Ophthalmol. (1)

G. Boehm, “Über Maculare (Haidingersche) Polarisations Büschel und Über einen Polarisation Optischen Fehler des Auges,” Acta Ophthalmol. 18, 109–142 (1940).
[CrossRef]

Am. J. Ophthalmol. (1)

D. S. Greenfield, R. W. Knighton, X.-R. Huang, “Effect of polarization axis on assessment of retinal nerve fiber layer thickness by scanning laser polarimetry,” Am. J. Ophthalmol. 129, 715–722 (2000).
[CrossRef] [PubMed]

Br. J. Ophthamol. (1)

A. Stanworth, E. J. Naylor, “The polarization optics of the isolated cornea,” Br. J. Ophthamol. 34, 201–211 (1950).
[CrossRef]

Cornea (1)

D. J. Donohue, B. J. Stoyanov, R. L. McCally, R. A. Farrell, “A numerical test of the normal incidence uniaxial model of corneal birefringence,” Cornea 5, 278–285 (1996).
[CrossRef]

Exp. Eye Res. (1)

W. A. Christens-Barry, W. J. Green, P. J. Connolly, R. A. Farrell, R. L. McCally, “Spatial mapping of polarized light transmission in the central rabbit cornea,” Exp. Eye Res. 62, 651–662 (1996).
[CrossRef] [PubMed]

Invest. Ophthalmol. Visual Sci. (4)

R. W. Knighton, X.-R. Huang, “Corneal compensation in scanning laser polarimetry: characterization and analysis,” Invest. Ophthalmol. Visual Sci. Suppl. 41, S92 (2000).

R. W. Knighton, X.-R. Huang, D. S. Greenfield, “Linear birerefringence measured in the central corneas of a normal population,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S131 (2001).

R. A. Farrell, R. L. McCally, D. Rouseff, “Corneal birefringence models,” Invest. Ophthalmol. Visual Sci. Suppl. 42, S281 (2001).

R. W. Knighton, X.-R. Huang, “Linear birefringence of the central human cornea,” Invest. Ophthalmol. Visual Sci. 43, 82–86 (2002).

J. Exp. Biol. (1)

A. Stanworth, E. J. Naylor, “Polarized light studies of the cornea. I. The isolated cornea,” J. Exp. Biol. 30, 160–163 (1953).

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

J. Physiol. (London) (1)

D. M. Maurice, “The structure and transparency of the corneal stroma,” J. Physiol. (London) 136, 263–285 (1957).

J. Refract. Surg. (1)

R. A. Farrell, J. F. Wharam, D. Kim, R. L. McCally, “Polarized light propagation in corneal lamellae,” J. Refract. Surg. 15, 700–705 (1999).
[PubMed]

Nature (1)

C. C. D. Shute, “Haidinger’s brushes and predominant orientation of collagen in corneal stroma,” Nature 250, 163–164 (1974).
[CrossRef] [PubMed]

Ophthalmic Physiol. Opt. (1)

J. W. Jaronski, H. T. Kasprzak, “Linear birefringence measurements of the in vitro human cornea,” Ophthalmic Physiol. Opt. 23, 361–369 (2003).
[CrossRef] [PubMed]

Physica (Amsterdam) (1)

H. L. D. Vries, A. Spoor, R. Gielof, “Properties of the eye with respect to polarized light,” Physica (Amsterdam) 19, 419–432 (1953).
[CrossRef]

Vision Res. (1)

L. J. Bour, N. J. Lopes Cardozo, “On the birefringence of the living human eye,” Vision Res. 21, 1413–1421 (1981).
[CrossRef] [PubMed]

Other (3)

D. Rouseff, R. A. Farrell, R. L. McCally, “Numerical modeling of the cornea’s birefringence properties. Nonnormal incidence effects,” manuscript available from the authors (russell.mccally@jhuap/.edu).

As used by Jones, S(ϕ) has the effect of rotating a vector counterclockwise through an angle ϕ. S(−ϕ) is the standard rotation matrix used to express a vector in a coordinate system rotated counterclockwise by an angle ϕ from the “laboratory system.”

This can be shown rigorously by examining the implications of setting θ=0 in Eq. (19). Doing so, and realizing that both of the squared terms must be zero in order for γ¯ to be zero, we obtain two equations for tan2ϕr as functions of ϕ3. We used Mathematica to show that the two equations do not have a common solution.

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Figures (7)

Fig. 1
Fig. 1

Illustration of the geometry of a wave incident at angle θ on a birefringent sheet that lies parallel to the x y plane. The optic axis is parallel to the x y plane and it is at an angle ϕ measured from the x axis. The plane of incidence, defined by the incident wave vector k and the z axis, is oriented at angle ϕ r measured from the x axis.

Fig. 2
Fig. 2

Incidence configurations for which the equivalent retardation of a two-plate system vanishes. The solid curve shows the angle of incidence θ γ ¯ = 0 as a function of the relative orientations of the two plates, ϕ 2 ϕ 1 (note ϕ 1 was set equal to zero without loss of generality). The dashed curve shows the orientation of the plane of incidence.

Fig. 3
Fig. 3

Incidence configurations for which the equivalent retardations of three-layer Sohncke systems (viz., ϕ 1 = 0 , ϕ 3 = 2 ϕ 2 ) vanish. (a) Bold solid curve, angle of incidence θ γ ¯ = 0 as a function of ϕ 2 for the case where 2 γ , the single plate retardation at normal incidence, is small; thin solid curve, orientation of the plane of incidence for this case; bold long-dashed curve, angle of incidence θ γ ¯ = 0 as a function of ϕ 2 for 2 γ = π 4 ; bold dotted curve, angle of incidence θ γ ¯ = 0 as a function of ϕ 2 for 2 γ = π 2 . In the latter two examples only the lowest-order solutions for θ γ ¯ = 0 are shown. (b) Results for 2 γ = π 4 . Bold dashed and dotted curves, higher-order solutions for θ γ ¯ = 0 . Thin solid curve, orientation of the plane of incidence for this case. (c) Results for 2 γ = π 2 . Bold dashed and dotted curves, higher-order solutions for θ γ ¯ = 0 ; thin solid curve, orientation of the plane of incidence for this case.

Fig. 4
Fig. 4

(a) Incidence configurations for which the equivalent rotations of three-layer Sohncke systems (viz., ϕ 1 = 0 , ϕ 3 = 2 ϕ 2 ) vanish shown as functions of ϕ 2 . The solid curve is for the case where 2 γ , the single plate retardation at normal incidence, is small. The long-dash curve is for the case where 2 γ = π 4 , and the dotted curve is for the case where 2 γ = π 2 . Each curve is calculated for the plane of incidence where the equivalent retardation also vanishes. (b)–(d) show how the angles of incidence for which the rotation and effective retardation vanish deviate from one another. The bold curves show the angle of incidence for which the rotation vanishes and the thin solid curves show the incidence angles where the effective retardation vanishes. (b) Case where 2 γ , the single plate retardation at normal incidence, is small. (c) Case where 2 γ = π 4 . (d) Case where 2 γ = π 2 .

Fig. 5
Fig. 5

Equivalent rotation, α ¯ , at the locations θ ( 2 k , γ ¯ = 0 ) and ϕ r = ϕ r γ ¯ = 0 (i.e., the locations where γ ¯ vanishes) for 2 γ = π 4 (dashed curve) and 2 γ = π 2 (solid curve). The curve is not shown for small γ because its maximum and minimum values are only ± 0.00153 ° at ϕ 2 = π 3 and 5 π 3 , respectively.

Fig. 6
Fig. 6

Solid curve is the angle of incidence for which the equivalent retardation of three-layer systems in which ϕ 1 = 0 and ϕ 2 = 2 ϕ 3 vanishes, shown as a function of ϕ 3 . The example shown is for the case where 2 γ , the single plate retardation at normal incidence, is equal to π 4 . The dashed curve is the orientation of the plane of incidence. It is to be noted that, unlike the case of the Sohncke systems in Fig. 3, this plane is not oriented in the direction of the plane of symmetry, ϕ r = ϕ 3 , given by the dotted line.

Fig. 7
Fig. 7

Incidence configurations for which the equivalent retardations of three-layer systems vanish. In these systems, 2 γ , the single plate retardation at normal incidence, is small, and the orientations of the first and third layers are fixed. the orientation of the second layer is varied. The solid curve is the incident angle for which the equivalent retardation vanishes as a function of the orientation of the second plate, ϕ 2 , and the dashed curve shows the orientation of the plane of incidence.

Equations (31)

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E out = M ( p ) E inc ,
M ( p ) = S ( ϕ ) G ( γ ) S ( ϕ ) ,
S ( ϕ ) = ( cos ϕ sin ϕ sin ϕ cos ϕ ) ,
G ( γ ) = ( exp ( i γ ) 0 0 exp ( i γ ) ) .
M ( N ) = M N ( p ) M N 1 ( p ) M 2 ( p ) M 1 ( p ) = ( m 11 ( N ) m 12 ( N ) m 21 ( N ) m 22 ( N ) ) ,
M ( N ) S ( ϕ ¯ ) G ( γ ¯ ) S ( ϕ ¯ ) S ( α ¯ ) .
m 11 ( N ) = cos α ¯ cos γ ¯ + i sin γ ¯ cos ( 2 ϕ ¯ α ¯ ) ,
m 21 ( N ) = sin α ¯ cos γ ¯ + i sin γ ¯ sin ( 2 ϕ ¯ α ¯ ) .
sin 2 γ ¯ = [ Im m 11 ( m ) ] 2 + [ Im m 21 ( N ) ] 2 ,
tan α ¯ = Re m 21 ( N ) Re m 11 ( N ) ,
tan ( 2 ϕ ¯ α ¯ ) = Im m 21 ( N ) Im m 11 ( N ) ,
2 ζ j = k e z , j k o z , j ,
ζ j = { γ j [ ( 1 + δ j ) 1 2 1 ] } ( { ( 1 + δ j ) sin 2 θ [ 1 + δ j cos 2 ( ϕ j ϕ r ) ] } 1 2 cos θ ) .
sin ψ j = sin ( ϕ j ϕ r ) [ 1 sin 2 θ cos 2 ( ϕ j ϕ r ) ] 1 2 ,
cos ψ j = cos θ cos ( ϕ j ϕ r ) [ 1 sin 2 θ cos 2 ( ϕ j ϕ r ) ] 1 2 ,
m 11 ( 1 ) = cos ζ + i sin ζ cos 2 ψ ,
m 21 ( 1 ) = i sin ζ sin 2 ψ .
m 11 ( 2 ) = cos ζ 1 cos ζ 2 sin ζ 1 sin ζ 2 cos 2 ( ψ 1 ψ 2 ) + i [ cos ζ 2 sin ζ 1 cos 2 ψ 1 + cos ζ 1 sin ζ 2 cos 2 ψ 2 ] ,
m 21 ( 2 ) = sin ζ 1 sin ζ 2 sin 2 ( ψ 2 ψ 1 ) + i [ cos ζ 2 sin ζ 1 sin 2 ψ 1 + cos ζ 1 sin ζ 2 sin 2 ψ 2 ] .
m 11 ( 3 ) = cos ζ 1 cos ζ 2 cos ζ 3 [ sin ζ 1 sin ζ 2 cos ζ 3 cos 2 ( ψ 2 ψ 1 ) + sin ζ 3 sin ζ 1 cos ζ 2 cos 2 ( ψ 3 ψ 1 ) + sin ζ 2 sin ζ 3 cos ζ 1 cos 2 ( ψ 3 ψ 2 ) ] + i [ cos ζ 3 cos ζ 2 sin ζ 1 cos 2 ψ 1 + cos ζ 1 cos ζ 3 sin ζ 2 cos 2 ψ 2 + cos ζ 2 cos ζ 1 sin ζ 3 cos 2 ψ 3 sin ζ 3 sin ζ 2 sin ζ 1 cos 2 ( ψ 3 ψ 2 + ψ 1 ) ] ,
m 21 ( 3 ) = [ sin ζ 1 sin ζ 2 cos ζ 3 sin 2 ( ψ 2 ψ 1 ) + sin ζ 3 sin ζ 1 cos ζ 2 sin 2 ( ψ 3 ψ 1 ) + sin ζ 2 sin ζ 3 cos ζ 1 sin 2 ( ψ 3 ψ 2 ) ] + i [ cos ζ 3 cos ζ 2 sin ζ 1 sin 2 ψ 1 + cos ζ 1 cos ζ 3 sin ζ 2 sin 2 ψ 2 + cos ζ 2 cos ζ 1 sin ζ 3 sin 2 ψ 3 sin ζ 3 sin ζ 2 sin ζ 1 sin 2 ( ψ 3 ψ 2 + ψ 1 ) ] .
sin 2 γ ¯ = sin 2 ( ζ 2 ζ 1 ) sin 2 ( ψ 2 ψ 1 ) + sin 2 ( ζ 2 + ζ 1 ) cos 2 ( ψ 2 ψ 1 ) ,
tan α ¯ = 2 sin ( ψ 2 ψ 1 ) cos ( ψ 2 ψ 1 ) [ cot ζ 2 cot ζ 1 cos 2 ( ψ 2 ψ 1 ) ] ,
tan ( 2 ϕ ¯ α ¯ ) = [ cos ζ 2 sin ζ 1 sin 2 ψ 1 + cos ζ 1 sin ζ 2 sin 2 ψ 2 ] [ cos ζ 2 sin ζ 1 cos 2 ψ 1 + cos ζ 1 sin ζ 2 cos 2 ψ 2 ] ,
sin 2 γ ¯ = { [ cos ζ j + 2 cos ζ j + 1 sin ζ j sin 2 ψ j ] sin ζ 3 sin ζ 2 sin ζ 1 sin 2 ( ψ 3 ψ 2 + ψ 1 ) } 2 + { [ cos ζ j + 2 cos ζ j + 1 sin ζ j cos 2 ψ j ] sin ζ 3 sin ζ 2 sin ζ 1 cos 2 ( ψ 3 ψ 2 + ψ 1 ) } 2 ,
cos 2 θ ( 2 k , γ ¯ = 0 ) = sin ( 2 ζ 1 ζ 2 ) tan 2 ϕ 2 sin ( 2 ζ 1 + ζ 2 )
cos 2 θ ( 2 k + 1 , γ ¯ = 0 ) = sin ( 2 ζ 1 + ζ 2 ) sin ( 2 ζ 1 ζ 2 ) tan 2 ϕ 2 ,
cos 2 θ ( 2 k , α ¯ = 0 ) = ( tan 2 ϕ 2 ) sin ( ζ 1 ζ 2 ) sin ( ζ 1 + ζ 2 ) ,
cos 2 θ ( 2 k + 1 , α ¯ = 0 ) = sin ( ζ 1 + ζ 2 ) ( tan 2 ϕ 2 ) sin ( ζ 1 ζ 2 ) .
tan ( 2 ϕ r γ ¯ = 0 ) = j = 1 3 sin ( 2 ϕ j ) j = 1 3 cos ( 2 ϕ j ) ,
cos 2 θ γ ¯ = 0 = j = 1 3 sin 2 ( ϕ j ϕ r γ ¯ = 0 ) j = 1 3 cos 2 ( ϕ j ϕ r γ ¯ = 0 ) .

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